Optimization of a Microchannel Heat Sink with Varying Channel Heights and Widths

The optimal tapered channel design of a microchannel heat sink is obtained by a combined optimization procedure that includes a model of a three-dimensional microchannel heat sink and a simplified conjugate-gradient method. The objective function to be minimized is the overall thermal resistance with the number of channels N, channel-width ratio β, height-tapered ratio Λ _y , and width-tapered ratio Λ _z as the design parameters. It is shown that the thermal resistance in all its relationships with the individual parameters exhibits a decrease followed by an increase. The thermal resistance is sensitive to the variations in the channel number, channel-width ratio, or width-tapered ratio while less sensitive to the height-tapered ratio. Optimization results show that for a given pumping power (0.5 W), the optimal design variables are N = 78, β = 0.78, Λ _z  = 0.59, and Λ _y  = 0.81 with a corresponding minimum overall thermal resistance of 0.087 K W^?1. These optimal design variables produce a 37.6% decrement in thermal resistance compared to the initial parallel-channel design estimate (N = 71, β = 0.85, Λ _z  = 0.99, and Λ _y  = 0.99). Additionally, as the pumping power increases, the optimal thermal resistance decreases and the corresponding optimal values of N increase; whereas, β, Λ _z , and Λ _y decrease.     

On the Validity of the Isothermal Model for Natural Convection Flow around Blocks Submitted to Volumetric Heat Generation in a Horizontal Channel

This work presents numerical results of natural convection in a horizontal channel provided with heating blocks periodically mounted on its lower adiabatic surface. The upper surface of the channel is maintained cold at a constant temperature. The parameters of the study are the ratio of solid blocks to fluid thermal conductivities (0.1 ≤ k* = k _s /k _a ≤ 200), the Rayleigh number (10⁴ ≤ Ra ≤ 10⁷), and the relative blocks height (1/8 ≤ B ≤ 1/2). Two models are considered in this study depending on whether the blocks are submitted to uniform heat generation (model 1), or maintained isothermal (model 2) at the average temperature calculated using model 1. The effect of the thermal conductivities ratio and the other controlling parameters on the validity of the isothermal model is examined. It is found that when multiple steady solutions are possible, some of the solutions obtained with the isothermal model may not reproduce the results of the model with blocks submitted to volumetric heat generation, even at very large conductivities ratio.     

Aspect ratio effect on natural convection in a square enclosure with a sinusoidal active thermal wall using a thermal non-equilibrium model

ABSTRACT

A numerical investigation of the aspect ratio effect on natural convection in a square enclosure is carried out by adopting the local thermal non-equilibrium model. The top and bottom walls of the enclosure are adiabatic, the left vertical wall is partially heated and cooled by the sinusoidal thermal boundary condition, and the right vertical wall is maintained at uniform thermal boundary condition. The results show the value of periodicity parameter increasing. The streamlines vary in different patterns, rotating clockwise and counterclockwise simultaneously when N > 1, and the number of clockwise and counterclockwise rotating cells increases with the increase of N and equals the value of N. The sinusoidal local Nusselt number profiles are observed and the wave amplitude of local Nusselt number decreases with the increase of aspect ratio, and the absolute values of average Nusselt number at left wall of porous cavity reach maximum when Ar = 1. The absolute value of solid-to-fluid temperature differences decreases as the inter-phase heat transfer coefficient (H) increases and it increases as the value of aspect ratio increases. The total heat transfer of porous cavity can be enhanced by increasing the aspect ratio and the thermal conductivity ratio.     

Natural Convection of a Binary Fluid in a Slightly Inclined Shallow Cavity

ABSTRACT

This article reports an analytical and numerical study of the natural convection in an inclined shallow cavity filled with a binary fluid. Newmann boundary conditions for temperature are applied to the long side walls of the enclosure, while the two short ones are assumed to be impermeable and insulated. The solutal buoyancy force are induced either by the imposition of constant fluxes of solute on the walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effects, a = 1). The governing parameters for the problem are the thermal Rayleigh number,Ra_T, the Lewis number Le, the buoyancy ratio ?, the inclination of the cavity Θ, the Prandtl numberPr, the aspect ratio of the cavity A, and the constant a. For convection in an infinite layer (A > > 1), an analytical solution of the steady form of the governing equations is obtained on the basis of the parallel flow approximation. The critical Rayleigh numbers for the onset of supercritical and subcritical convection are predicted by the present model. Also, it is demonstrated that, for small enough inclinations around the horizontal plane, multiple steady states exist, some of which are unstable. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. Good agreement is observed between the analytical prediction and the numerical simulations.     

Nonhomogeneous Heat Conduction Problem and Its Thermal Deflection Due to Internal Heat Generation in a Thin Hollow Circular Disk

This article is concerned with the determination of temperature and thermal deflection in a thin hollow circular disk under an unsteady-state temperature field due to internal heat generation within it. Initially, the disk is kept at an arbitrary temperature F(r, z). For times t > 0 heat is generated within the thin hollow circular disk at a rate of g(r, z, t) Btu/hr ft³, while the boundary surfaces at (r = a), (r = b), (z = 0) and (z = h) are kept at temperatures f ₁(z, t) and f ₂(z, t), f ₃(r, t) and f ₄(r, t), respectively. The governing heat conduction equation has been solved by using a finite Hankel transform and the generalized finite Fourier transform. The results are obtained in series form in terms of Bessel's functions. As a special case, different metallic disks have been considered. The results for temperature change and the thermal deflection have been computed numerically and illustrated graphically.     

Numerical Modeling of the Freezing of a Porous Humid Food inside a Cavity due to Natural Convection

The conjugate heat transfer problem of food freezing inside a cavity was numerically investigated. A vegetable sponge has been considered as a food block freezing inside a square freezing chamber due to natural convection. The need for specifying the block surface convective heat transfer coefficients was eliminated by solving the surrounding cooling fluid and, therefore, a comprehensive understanding on heat transfer and airflow during freezing can be achieved. The 2-D unsteady Navier-Stokes and energy equations were solved using a finite volume method with the semi-implicit SIMPLE algorithm. Thermophysical properties of food block components were considered to be dependent on temperature, as well as moisture and ice content, which has been rarely considered in food freezing studies. The specific heat capacity method was employed to model the freezing process. The Krischer model was adopted for predicting the thermal conductivity of the food, which indicates that the model works with reasonable accuracy for humid porous foods. The mechanism of natural convection in the cavity was carefully studied and the effects of different parameters on the freezing time were examined. The food freezing curves were investigated for various Rayleigh numbers in the range of 10⁴ ≤ Ra ≤ 10⁶, with different area ratios of A = 1/16, 1/9, and 1/4, and food initial water contents of X _tw  = 0.48, 0.58, and 0.68. It was concluded that increasing the Rayleigh number reduces the freezing time. On the other hand, area ratio and initial water content of the food were proved to extend the freezing time.     

Numerical investigation on heat transfer of supercritical water in a roughened tube

ABSTRACT

The turbulent mixed convection heat transfer of supercritical water flowing in a vertical tube roughened by V-shaped grooves has been numerically investigated in this paper. The turbulent supercritical water flow characteristics within different grooves are obtained using a validated low-Reynolds number κ-ε turbulence model. The effects of groove angle, groove depth, groove pitch-to-depth ratio, and thermophysical properties on turbulent flow and heat transfer of supercritical water are discussed. The results show that a groove angle γ = 120° presents the best heat transfer performance among the three groove angles. The lower groove depth and higher groove pitch-to-depth ratio suppress the enhancement of heat transfer. Heat transfer performance is significantly decreased due to the strong buoyancy force at T_b = 650.6 K, and heat transfer deterioration occurs in the roughened tube with γ = 120°, e = 0.5 mm, and p/e = 8 in the present simulation. The results also show that the rapid variation in the supercritical water property in the region near the pseudo-critical temperature results in a significant enhancement of heat transfer performance.     

Magneto-Convective Transport of Nanofluid in a Vertical Lid-Driven Cavity Including a Heat-Conducting Rotating Circular Cylinder

Numerical simulations are performed for the two-dimensional magneto-convective transport of Cu–H₂O nanofluid in a vertical lid-driven square cavity in the presence of a heat-conducting and rotating circular cylinder. The left wall of the cavity is allowed to translate at a constant velocity in the vertically upward direction. Both left and right walls are maintained at isothermal but different temperatures. The top and bottom walls of the enclosure are thermally insulated. At the central region of the cavity is a heat-conducting circular cylinder which can rotate either clockwise or counterclockwise. A constant horizontal magnetic field of amplitude B₀ is applied perpendicular to the vertical walls. The nanofluid is electrically conducting, while the solid walls are considered electrically insulated. Simulations are performed for various controlling parameters, such as Richardson number (0.01 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), dimensionless rotational speed of the cylinder (Ω = ±1), and nanoparticle concentration (0 ≤ ? ≤ 0.3), while Reynolds number based on lid velocity is fixed at a specific value (Re = 100). The flow and thermal fields are found to be susceptible to changes in the magnetic field and mixed convective strength, as well as nanoparticle concentration. However, the direction of cylinder rotation is observed to have little or no influence quantitatively on global hydrodynamic and thermal parameters.     

Coupled Natural Convection and Radiation in a Horizontal Rectangular Enclosure Discretely Heated from Below

A numerical study is carried out to investigate the interaction between natural convection and thermal radiation in a horizontal enclosure filled with air and heated discretely from below. The results are presented for a cavity having an aspect ratio A _r  = L′/H′ = 10, while the Rayleigh number and the emissivity of the walls are varied in the ranges 10³ ≤ Ra ≤ 10⁶ and 0 ≤ ε ≤ 1, respectively. The results of the study, presented in terms of flow and temperature patterns, average convective, radiative and total Nusselt numbers, evaluated on the cold wall, show that the problem has multiple solutions. Each of these solutions is characterized by a specific flow structure, and its appearance and range of existence depend strongly on the parameters Ra and ε. The amount of heat evacuated through the cold surface is dependent on the type of solution.     

NUMERICAL SHAPE OPTIMIZATION FOR HIGH PERFORMANCE OF A HEAT SINK WITH PIN-FINS

The design optimization of a 7 × 7 pin-fin heat sink is performed numerically. To achieve higher thermal performance of the heat sink, the thermal resistance at the junction of the chip and the heat sink and the overall pressure drop in the heat sink have to be minimized simultaneously. The fin height (h), fin width (w), and fan-to-heat sink distance (c) are chosen as the design variables, and the pressure drop (ΔP) and thermal resistance (θ _ja ) are adopted as the objective functions. To obtain the optimum design values, we used the finite-volume method for calculating the objective functions, the Broydon-Fletcher-Goldfarb-Shanno method for solving the unconstrained nonlinear optimization problem, and the weighting method for predicting the multiobjective problem. The results show that the optimum design variables for the weighting coefficient of 0.5 are as follows: w = 4.653 mm, h = 59.215 mm, and c = 2.669 mm. The objective functions corresponding to the optimal design are calculated as ΔP = 6.82 Pa and θ _ja  = 0.56 K/W. The Pareto solutions are also presented for various weighting coefficients, and they offer very useful data for designing a pin-fin heat sink.     

Effects of Wall-Located Heat Barrier on Conjugate Conduction/Natural-Convection Heat Transfer and Fluid Flow in Enclosures

The effects of a heat barrier, located in the ceiling wall of an enclosure, on conjugate conduction/natural convection are investigated numerically. The vertical walls of the enclosure are differentially heated and the horizontal walls are adiabatic. Heatline technique is used to visualize heat transport. The variations of average Nusselt number, dimensionless heat transfer rate through the ceiling wall, and dimensionless overall heat transfer rate are studied. Calculations are performed for different Rayleigh numbers (10³ ≤ Ra ≤ 10⁶), thermal conductivity ratios (1 ≤ K ≤ 100), dimensionless locations of the heat barrier (0 < X _h  < 1),and two dimensionless ceiling wall thicknesses (D = 0.05 and D = 0.20). For high thermal conductivity ratio (K = 100), the heat barrier considerably reduces the dimensionless overall heat transfer rate. The effect of the heat barrier on dimensionless heat transfer rate through the enclosure increases as the Rayleigh number decreases. For low Rayleigh number (i.e., Ra = 10³), a location exists in the ceiling wall for which the dimensionless overall heat transfer rate is minimum.     

Simulation of natural convection and entropy generation of MHD non-Newtonian nanofluid in a cavity using Buongiorno's mathematical model

In this paper, natural convection and entropy generation of non-Newtonian nanofluid, using the Buongiorno's mathematical model in a cavity in the presence of a uniform magnetic field has been analyzed by Finite Difference Lattice Boltzmann method (FDLBM). The cavity is filled with nanofluid which the mixture shows shear-thinning behavior. This study has been performed for the certain pertinent parameters of Rayleigh number (Ra = 10⁴ and 10⁵), Hartmann number (Ha = 0, 15, 30), buoyancy ratio number (N_r = 0.1, 1, and 4), power-law index (n = 0.4–1), Lewis number (Le = 1, 5, and 10), Thermophoresis parameter (N_t = 0.1, 0.5, 1), and Brownian motion parameter (N_b = 0.1, 1, 5). The Prandtl number is fixed at Pr = 1. The Results indicate that the augmentation of Hartmann number causes heat and mass transfer to drop. The increase in Rayleigh number enhances heat and mass transfer for various power-law indexes. The alteration of the power-law index changes heat and mass transfer. In addition, the rise of Hartmann number declines the shear-thinning behavior. The increase in the Lewis number augments mass transfer while it causes heat transfer to drop. The rise of the Thermophoresis and Brownian motion parameters ameliorate mass transfer and declines heat transfer significantly. The augmentation of buoyancy ratio number enhances heat and mass transfer. The augmentation of the power-law index declines various entropy generations in different Rayleigh numbers and Hartmann numbers. The increase in Hartmann number declines total entropy generation in different Rayleigh numbers. In addition, the rise of Rayleigh number and Hartmann number causes Bejan number to drop in various power-law indexes. The enhancement of the Lewis number provokes the total irreversibility to rise. Further, the total entropy generation increases as the buoyancy ratio number augments. It was shown that the increase in the Brownian motion and Thermophoresis parameters enhance the total irreversibility.     

OPTIMUM DESIGN OF PLATE HEAT EXCHANGER WITH STAGGERED PIN ARRAYS

The optimization of the plate heat exchanger with staggered pin arrays for a fixed volume is performed numerically. The flow and thermal fields are assumed to be periodic fully developed flow and heat transfer with constant wall temperature and they are solved using the finite-volume method. The optimization is carried out by using the sequential linear programming (SLP) method, and the weighting method is adopted for solving the multiobjective problem. The results show that the optimal design variables for the weighting coefficient of 0.5 are as follows: S = 6.497 mm, P = 5.496 mm, D ₁ = 0.689 mm, and D ₂ = 2.396 mm. The Pareto optimal solutions are presented also.     

Anisotropy Effects on Heat and Fluid Flow by Unsteady Natural Convection and Radiation in Saturated Porous Media

ABSTRACT

This article deals with a numerical study of fluid flow and heat transfer by unsteady natural convection and thermal radiation in a vertical channel opened at both ends and filled with anisotropic, in both thermal conductivity and permeability, fluid-saturated porous medium. The bounding walls of the channel are gray and kept at a constant hot temperature.

In the present study we suppose the validity of the Darcy law for motion and of the local thermal equilibrium assumption. The radiative transfer equation (RTE) is solved by the finite-volume method (FVM). The numerical results allow us to represent the time–space variations of the different state variables. The sensitivity of the fluid flow and the heat transfer to different controlling parameters, namely, the single scattering albedo ω, the temperature ratio R, the anisotropic thermal conductivity ratio R_c, and the anisotropic permeability ratio R_k, are addressed. Numerical results indicate that the controlling parameters of the problem, namely, ω, R, R_c, and R_k, have significant effects on the flow and thermal field behavior and also on the transient process of heating or cooling of the medium. Effects of such parameters on time variations of the volumetric flow rate q_v and the convected heat flux Q at the channel's outlet are also studied.     

Computations of Newtonian fluid flow around a square cylinder near an adiabatic wall at low and intermediate Reynolds numbers: Effects of cross-buoyancy mixed convection

ABSTRACT

The effects of cross-buoyancy mixed convection from a square cylinder in the proximity of a plane wall are studied for Reynolds number (Re) = 1–100, Richardson number (Ri) = 0–2, and gap ratio (G) = 0.25–1 at Prandtl number (Pr) = 0.7. The flow observed is steady for G = 0.25 and 0.5. The transition from a steady to a time-periodic system is observed for G = 1, and it is found at Re = 56, 60, and 74 for Ri = 0, 1, and 2, respectively. With increasing G and/or Ri, the drag coefficient and average Nusselt number increase for all Re values studied and the lift coefficient decreases with increasing Ri except at Re = 1. Maximum heat transfer augmentation is found about 89% at G = 0.5 (Re = 20, Pr = 0.7, Ri = 0) with respect to the corresponding value at G = 0.25 (Re = 20, Pr = 0.7, Ri = 0). Lastly, the correlations of drag coefficient and heat transfer have been obtained.     

A thermodynamic analysis of biogas partial oxidation to synthesis gas with emphasis on soot formation

A thermodynamic analysis of synthesis gas production via partial oxidation (POX) of biogas is performed in the present article. Chemical equilibrium calculations are conducted for partial oxidation of (CH₄+CO₂) mixtures based on Gibbs free energy minimization method emphasizing soot formation. Regarding precise evaluation of carbon dioxide effects on the reforming characteristics, the obtained results are compared with the experimental data. Furthermore, the effects of steam injection at the inlet of the reformer on the coking behavior and syngas production yield are studied. To investigate the effects of the equivalence ratio (?), temperature and pressure, a broad parametric study is performed. The results reveal that the process temperature plays a pivotal role in enhancing the syngas production and soot abatement. It is also found that the pressure has an impractical effect on the syngas production yield, leading to the soot formation and decrease in both hydrogen and carbon monoxide yields. Furthermore, increasing the inlet CO₂/CH₄ makes the thermal reforming efficiency to rise at an equivalence ratio lower than 3. Meanwhile, increasing the steam to methane (S/C) ratio reduces carbon formation and enhances hydrogen production. Nonetheless, when the S/C ratio is larger than 2 at ? = 2.5 and 1 at ? = 3, the enhancement of hydrogen generation is minimized and even tends to become impractical. Therefore, near adiabatic and atmospheric condition at ? = 2.5–3 with S/C < 1 are recommended as the optimum operating routes for partial oxidation of biogas.     

Numerical study on the melting thermal characteristics of a microencapsulated phase change plate

ABSTRACT

The melting thermal characteristics of a microencapsulated phase change material (MEPCM) plate under constant heat flux are numerically investigated. The effective property relations of the MEPCM plate related to the parameters of base materials are first established and verified by the measured thermal properties of 10 samples. A one-dimensional phase change thermal model based on the enthalpy method is built, and the predicted results are also verified with test data. The temperature profile and phase interface movement in the MEPCM plate are discussed, and the effects of Stefan number, phase change temperature range, particle and core fraction, and additive fraction are also analyzed. Three states of solid, mushy, and fluid phases of phase change material (PCM) core divide the melting process of the MEPCM plate into five stages as follows: Fo ≤ 0.1, 0.1 < Fo ≤ 0.2, 0.2 < Fo ≤ 0.9, 0.9 < Fo ≤ 1.2, and Fo ≥ 1.2. The time point of regular regime for heat transfer in the MEPCM plate is Fo = 0.2. The addition of conductivity additive homogenizes the temperature profile in the MEPCM plate. Under the combined effects of thermal diffusivity and Ste number, an optimum additive fraction exists for the MEPCM plate to achieve the minimum melting completion time or the maximum latent storage efficiency.     

Natural Convection: Analysis of Partially Open Enclosures With an Internal Heated Source

A numerical study of the thermal and fluid dynamic behavior of air in partially open two-dimensional enclosures is presented. An analysis is made based on two aspects of the radius, H/W = 1 and 2. The left and right walls are maintained at different constant temperatures, while the upper and bottom walls are thermally insulated. The enclosure has an opening on the right wall and a small heating source located on the bottom or left vertical wall, occupying three different positions. Numerical simulations were performed for several values of Rayleigh number (Ra _e ) in the range between 10³ and 10⁶>, while the intensity of the two effects—the difference in temperature of the vertical walls and the internal heating source (Ra _i )—was evaluated based on the relation R = Ra _i /Ra _e , in the range between 0 and 2,500. Representative results illustrating the effects of relation R on the streamlines and isotherms within the enclosures are reported. In addition, simulation results for the local and average Nusselt numbers on the heated and colded walls of the enclosures are presented and discussed for different values of the parameters R, Ra _e , W _H , and H/W. It is founded that the parameter modifications have significant effects on the average and local Nusselt numbers of the enclosures.     

Entropy Generation During Natural Convection in a Porous Cavity: Effect of Thermal Boundary Conditions

Entropy generation plays a significant role in the overall efficiency of a given system, and a judicious choice of optimal boundary conditions can be made based on a knowledge of entropy generation. Five different boundary conditions are considered and their effect of the permeability of the porous medium, heat transfer regime (conduction and convection) on entropy generation due to heat transfer, and fluid friction irreversibilities are investigated in detail for molten metals (Pr = 0.026) and aqueous solutions (Pr = 10), with Darcy numbers (Da) between 10^?5–10^?3 and at a representative high Rayleigh number, Ra = 5 × 10⁵. It is observed that the entropy generation rates are reduced in sinusoidal heating (case 2) when compared to that for uniform heating (case 1), with a penalty on thermal mixing. Finally, the analysis of total entropy generation due to variation in Da and thermal mixing and temperature uniformity indicates that, there exists an intermediate Da for optimal values of entropy generation, thermal mixing, and temperature uniformity.     

The effects of geometric parameters on the thermal performance of a rectangular natural circulation loop containing PCM suspensions

ABSTRACT

A numerical study that uses the finite difference method to model the fluid flow and heat transfer of a rectangular natural circulation loop that contains phase change material (PCM) suspensions is presented to investigate how geometric parameters affect the thermal performance. Parametric simulations were performed using different geometrical parameters in the following ranges: the dimensionless length of the heated section = 0.4–1; the relative elevation of the cooled section compared with the heated section = 0.5–2; and the aspect ratio of the loop = 0.25–1. The results determine the important geometric parameters that affect the heat transfer performance of the loop with the PCM suspension. In several of the geometric configurations, the heat transfer performance of the loop is significantly affected by the latent heat contribution associated with the melting/freezing of the PCM particles.